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We remark that the a **posteriori** **error** **estimator** ¯ θ is locally efficient **in** the interior elements (that is, those that do not touch the boundary). Besides, the computation of ¯ θ involves four residuals per **element** **in** the interior tri- angles, five residuals per **element** **in** the triangles with exactly one vertex on the boundary and six residuals per **element** **in** the triangles with a side on the boundary. We emphasize that ¯ θ has been derived **in** the two-dimensional set- ting, assuming that displacements are approximated by continuous piecewise **linear** **finite** elements. **In** the next section, we provide some numerical results illustrating the performance of the corresponding adaptive **method**, including a numerical comparison with the a **posteriori** **error** **estimator** proposed **in** [4]. 4. Numerical experiments

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We present and analyse a new **mixed** **finite** **element** **method** **for** the generalized Stokes prob- lem. The approach, which is a natural extension of a previous procedure applied to quasi- Newtonian Stokes flows, is **based** on the introduction of the flux and the tensor gradient of the velocity as further unknowns. This yields a two-fold saddle point operator equation as the resulting variational formulation. Then, applying a slight generalization of the well known Babuˇ ska-Brezzi theory, we prove that the continuous and discrete formulations are well posed, and derive the associated a priori **error** analysis. **In** particular, the **finite** **element** subspaces providing stability coincide with those employed **for** the usual Stokes flows except **for** one of them that needs to be suitably enriched. We also develop **an** a-**posteriori** **error** estimate (**based** on local problems) and propose the associated adaptive algorithm to com- pute the **finite** **element** solutions. Several numerical results illustrate the performance of the **method** and its capability to localize boundary layers, inner layers, and singularities.

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Deformable Image Registration (DIR) is a powerful computational **method** **for** image analysis, with promising applications **in** the diagnosis of human disease. Despite being widely used **in** the medical imaging community, the mathematical and numerical analysis of DIR methods still has many open questions. Further, recent applications of DIR **in**- clude the quantification of mechanical quantities **in** addition to the aligning transformation, which justifies the development of novel DIR formulations **for** which the accuracy and convergence of fields other than the aligning transformation can be studied. **In** this work we propose and analyze a primal, **mixed** and **augmented** formulations **for** the DIR prob- lem, together with their **finite**-**element** discretization schemes **for** their numerical solution. The DIR variational problem is equivalent to the **linear** **elasticity** problem with a source term that has a nonlinear dependence on the unknown field. Fixed point arguments and small data assumptions are employed to derive the well-posedness of both the continuous and discrete schemes **for** the usual primal and **mixed** variational formulations, as well as **for** **an** **augmented** version of the later. **In** particular, continuous piecewise **linear** elements **for** the displacement **in** the case of the primal **method**, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) **for** the stress together with piecewise constants (resp. continuous piecewise **linear**) **for** the displacement when using the **mixed** approach (resp. its **augmented** version), constitute feasible choices that guarantee the stability of the asso- ciated Galerkin systems. A priori **error** estimates derived by using Strang-type lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of the **method**.

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On the other hand, we recall that the application of adaptive algorithms, **based** on a **posteriori** **error** estimates, usually guarantees the quasi-optimal rate of convergence of the ﬁnite **element** solution to boundary value problems. **In** addition, this adaptivity is specially necessary **for** nonlinear problems where no a priori hints on how to build suitable meshes are available. To this respect, we have shown recently that the combination of the usual Bank–Weiser approach from [1] with the analysis from [3,4] allows to derive fully explicit and reliable a **posteriori** **error** estimates **for** the dual-**mixed** variational formulations (showing a two- fold saddle point structure) of some **linear** and nonlinear problems (see, e.g. [2,6,7]). However, no a pos- teriori **error** analysis has been developed yet **for** the nonlinear Stokes problems studied **in** [5]. Therefore, as a natural continuation of our results **in** [5], **in** the present paper we apply the Bank–Weiser type a **posteriori** **error** analysis mentioned above to derive reliable estimates **for** the **mixed** ﬁnite **element** scheme (1.6). The rest of this work is organized as follows. **In** Section 2 we collect some basic results on Sobolev spaces and state the main result of this paper. The proof of our a **posteriori** estimate, which makes use of the Ritz projection of the **error**, is provided **in** Section 3. **In** Section 4 we prove the quasi-eﬃciency of the **estimator** and discuss on suitable choices **for** the auxiliary functions needed **for** its computation. Finally, several numerical results illustrating the good performance of the adaptive algorithm are reported **in** Section 5.

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Deformable image registration (DIR) represent a powerful computational **method** **for** image analy- sis, with promising applications **in** the diagnosis of human disease. Despite being widely used **in** the medical imaging community, the mathematical and numerical analysis of DIR methods remain un- derstudied. Further, recent applications of DIR include the quantification of mechanical quantities apart from the aligning transformation, which justifies the development of novel DIR formulations where the accuracy and convergence of fields other than the aligning transformation can be stud- ied. **In** this work we propose and analyze a primal, **mixed** and **augmented** formulations **for** the DIR problem, together with their **finite**-**element** discretization schemes **for** their numerical solution. The DIR variational problem is equivalent to the **linear** **elasticity** problem with a nonlinear source term that depends on the unknown field. Fixed point arguments and small data assumptions are em- ployed to derive the well-posedness of both the continuous and discrete schemes **for** the usual primal and **mixed** variational formulations, as well as **for** **an** **augmented** version of the later. **In** particular, continuous piecewise **linear** elements **for** the displacement **in** the case of the primal **method**, and Brezzi-Douglas-Marini of order 1 (resp. Raviart-Thomas of order 0) **for** the stress together with piecewise constants (resp. continuous piecewise **linear**) **for** the displacement when using the **mixed** approach (resp. its **augmented** version), constitute feasible choices that guarantee the stability of the associated Galerkin systems. A-priori **error** estimates derived by using Strang-type Lemmas, and their associated rates of convergence depending on the corresponding approximation properties are also provided. Numerical convergence tests and DIR examples are included to demonstrate the applicability of the **method**.

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27 Lee mas

We extend the applicability of the **augmented** dual-**mixed** **method** introduced recently **in** [4, 5] to the problem of **linear** **elasticity** with **mixed** boundary conditions. The **method** is **based** on the Hellinger–Reissner principle and the symmetry of the stress tensor is imposed **in** a weak sense. The Neuman boundary condition is prescribed **in** the **finite** **element** space. Then, suitable Galerkin least-squares type terms are added **in** order to obtain **an** **augmented** variational formulation which is coercive **in** the whole space. This allows to use any **finite** **element** subspaces to approximate the displacement, the Cauchy stress tensor and the rotation.

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We develop **an** a **posteriori** **error** analysis of **residual** type of a stabilized **mixed** **finite** **element** **method** **for** Darcy flow. The stabilized formulation is obtained by adding to the standard dual- **mixed** approach suitable **residual** type terms arising from Darcy’s law and the mass conservation equation. We derive sufficient conditions on the stabilization parameters that guarantee that the **augmented** variational formulation and the corresponding Galerkin scheme are well-posed. Then, we obtain a simple a **posteriori** **error** **estimator** and prove that it is reliable and locally efficient. Finally, we provide several numerical experiments that illustrate the theoretical results and support the use of the corresponding adaptive algorithm **in** practice.

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The approach there is **based** on the introduction of suitable Galerkin least-squares terms that arise from the constitutive and equilibrium equations, and from the relation defining the rotation **in** terms of the displacement. **In** [18], besides these Galerkin least- squares terms, a consistency term related with the non-homogeneous Dirichlet boundary condition is added. **In** the case of pure Dirichlet boundary conditions, the bilinear form of the **augmented** formulation is bounded and coercive on the whole space and hence, the associated Galerkin scheme is well-posed **for** any **finite** **element** subspace. Thus, it is possible to use as **finite** **element** subspaces some non-feasible choices **for** the usual (non-**augmented**) dual-**mixed** formulation. **In** particular, it is possible to employ Raviart- Thomas elements of lowest order to approximate the stress tensor, continuous piecewise **linear** elements **for** the displacement, and piecewise constants **for** the rotation. **In** the case of **mixed** boundary conditions, the trace of the displacement on the Neumann boundary can be approximated by continuous piecewise **linear** elements on **an** independent partition of that boundary whose mesh size needs to satisfy a compatibility condition with the mesh size of the triangulation of the domain.

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The development of the space-time **method** proceeds by considering a partition M of the total time interval, t ∈ J = (0, T ), 0 < T < ∞ , into N time steps { I n } N n=0 given by I n = { (t n , t n+1 ) } N n=0 . The length of the variable time step is given by ∆t n = t n+1 − t n . Using this notation, Q n = Ω × I n , are the nth space-time slabs. Within each space-time slab, the spatial domain is subdivided into D n elements. We define the jump operator across space-time slabs as,

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Almeida (Editors); Escola Politécnica da Universidade de Sao Paulo, Sao Paulo, Brasil.[r]

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the velocity and temperature equations and LPS grad-div stabilization. **In** [5] the weak convergence of the **method** is obtained but no **error** bounds **for** the **method** are proved. **In** particular, since no bounds are proved **for** the rate of convergence of the **method** the question of the dependence on the constants **in** the bounds on the Rayleigh numbers is not considered. Our aim **in** the present paper is, on the one hand, proving **error** bounds **for** the methods with constants that do not deteriorate **for** increasing values of the Rayleigh numbers and, on the other hand, doing this with the fewest possible stabilization terms. As a consequence, **error** bounds are obtained with constants independent on inverse powers of ν and κ **in** formulation (2) with only grad-div stabilization.

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From the two above examples it can be deducted that the FEM allows to define unequivocally the curve of the road axis, subject to be continuous c2 and minimizing a functional of the type (1). **In** this way, it is not ne cessary to use the traditional special curves such as straight lines, circles and clothoides, and therefore a more wide freedom **for** the road designer is reached. The axis can be defined using the FEM simply by the n £ de coordinates of some special points (joints) and so- me extra constraints **in** the slope and/or curvature. The coordinates of intermediate points between two con secutive joints can be obtained from the use of the shape or interpolation functions (6).

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Abstract-The solution to the problem of finding the optimum mesh design in the finite element method with the restriction of a given number of degrees of freedom, is [r]

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This system of equations are solved by use of **an** incremental formulation. Here ˆ u and ˆ p are the vectors of nodal point incremental displacements and pressures, respectively. t+∆t R is the vector of externally applied nodal point loads corresponding to time t + ∆t; and F = t FU, t FP T is the vector of nodal point forces corresponding to the internal **element** stresses at time t. Due to this, the subtraction at the right hand side is known as the out-of-balance force vector and may reach **an** small value at the end of each load increment. Generally, this value is established as a percentage of the first out-of-balance vector magnitude at a load increment.

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•We have obtained a numerical solution of a 1D nonlinear reaction-diffusion type model, representing the initial stages of atherosclerosis, which is a variant of the one prop[r]

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The use of a software **based** on the **finite** **element** **method**, is a good support **for** the realization of engineering designs that involve the determination of stress, displacement and strain **in** complex parts, especially with stress concentrators. Also, the software allows to determine the distribution of tensions **in** zones with concentrators of stresses with geometric form not defined analytically.

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It is very frequent **in** scientific literature to use the dam-break problem **for** the validation of a model. A domain of 5x200 m without friction at the bottom has been considered, ensuring depths of 1 m and 0.1 m as initial conditions **in** the two separated areas. Again the water depth **in** the axis of symmetry, at time t = 25 s, is presented. The ∆t used (0.2 s) is the maximum one **in** which there are no oscillations, so is equivalent to Courant´s number **for** one dimen- sion. Results match very well the exact solution as can be seen **in** Burguete & García-Navarro (2000).

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